Existence and uniqueness of $$E_{\infty }$$ E ∞ structures on motivic $$K$$ K -theory spectra

Abstract

We show that algebraic \({\textit{K}}\)-theory \(\mathsf {KGL}\), the motivic Adams summand \(\mathsf {ML}\) and their connective covers acquire unique \(E_{\infty }\) structures refining naive multiplicative structures in the motivic stable homotopy category. The proofs combine \(\Gamma \)-homology computations and work due to Robinson giving rise to motivic obstruction theory. As an application we employ a motivic to simplicial delooping argument to show a uniqueness result for \(E_\infty \) structures on the \(K\)-theory Nisnevich presheaf of spectra.